# Multi-Addressed Fiber Bragg Structures for Microwave-Photonic Sensor Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Addressed Fiber Bragg Structure

## 3. Mathematical Model

_{1}, ω

_{2}, and ω

_{3}are the frequencies of optical carriers; Ω

_{21}and Ω

_{32}are the address frequencies; k and b are the predefined parameters of the optic filter with an oblique amplitude–frequency characteristic.

_{21}, Ω

_{32}, and their sum Ω

_{31}= Ω

_{21}+ Ω

_{32}. The task is to determine the MAFBS central frequency (any of the frequencies ω

_{1}, ω

_{2}, or ω

_{3}), using the known parameters of the beating at the address frequencies set.

_{1}, A

_{2}, and A

_{3}are the amplitudes, φ

_{1}, φ

_{2}, and φ

_{3}are the initial phases of the signal at the optical frequencies ω

_{1}, ω

_{2}, and ω

_{3}. The output current of the photodetector F(t) is proportional to the square of the optic response:

_{21}and Ω

_{32}. The constant signal level in (2), the amplitudes at the address frequencies Ω

_{21}, Ω

_{32}, and their sum Ω

_{31}give four independent equations for the determination of three unknown amplitudes, A

_{1}, A

_{2}, and A

_{3}:

_{0}, D

_{21}, D

_{32}, and D

_{31}are the measured values of constant signal level, amplitudes of the address frequencies Ω

_{21}, Ω

_{32}, and their sum Ω

_{31}, respectively.

_{1}, A

_{1}), (ω

_{2}, A

_{2}), and (ω

_{3}, A

_{3}) belong to the same line:

_{2}− ω

_{1}and ω

_{3}− ω

_{2}are equal to the address frequencies Ω

_{21}and Ω

_{32}, respectively, and the condition ω

_{3}− ω

_{1}= Ω

_{32}+ Ω

_{21}would also be automatically satisfied. Thus, it is necessary to add the additional relation to Equation (3):

_{1}, A

_{2}, and A

_{3}from Equation (3), supplemented by Equation (5), and using the known values of the parameters k and b of the filter with an oblique linear amplitude–frequency characteristic, one can calculate the MAFBS frequencies ω

_{1}, ω

_{2}, and ω

_{3}.

_{1}, A

_{2}, A

_{3}, and λ are equal to zero, which leads to a set of four non-linear equations:

_{i}are not expressed due to their obvious simplicity.

_{i}, i = 1, 2, 3) can be solved only numerically. The values A

_{1}, A

_{2}, and A

_{3}, which are the solution of Equation (3) with the exception of the first equation, can be taken as the initial conditions, and λ can be taken as the initial value equal to zero:

_{1}, A

_{2}, and A

_{3}, each of which can be used to determine the MAFBS central frequency position relative to the filter with an oblique amplitude–frequency characteristic. Substituting the found values of the amplitudes A

_{1}, A

_{2}, and A

_{3}in Equation (4), and combining them in Equation (6), we obtain the expression for the central frequency of the MAFBS:

_{0}, D

_{21}, D

_{32}, and D

_{31}—a constant signal level, and amplitudes at address frequencies Ω

_{21}, Ω

_{32}, and their sum Ω

_{31}, respectively.

## 4. Generalized Modulation Factor

## 5. Results of Numerical Modeling

^{0}and dimensional amplitude A

^{0}. As the characteristic dimensional frequency of task Ω

^{0}, we set the frequency corresponding to 125 GHz (in wavelength terms, it is 1 nm). The characteristic dimensional amplitude A

^{0}depends on the maximum output current of the photodetector, which can be independently amplified or attenuated to any value. We normalize all the task variables, so that the maximum central frequency shifting of the MAFBS does not exceed 125 GHz or Ω

^{0}, which is equal to 1 conventional unit. We normalize the maximum signal amplitude, so that in dimensionless quantities the maximum signal level does not exceed 1000 conventional units. Based on this, we define the parameters of optic filter with an oblique linear amplitude–frequency characteristic as k = 1000 conventional units and b = 100 conventional units. As the MAFBS model, we choose a structure with address frequencies Ω

_{21}= 0.01 conventional units (1.25 GHz) and Ω

_{32}= 0.02 conventional units (2.50 GHz), with a range of changes in the MAFBS central frequency to 1 conventional unit.

_{F}of determining the amplitudes D

_{0}, D

_{21}, D

_{32}and D

_{31}.

_{F}of 0.01% and 0.001% of the full scale. The dependence of the modulation factor on the central frequency shift of the MAFBS at amplitude determination error equal to 0.01% is shown in Figure 3a, and at 0.001% is shown in Figure 3b. The blue line indicates the modulation factor, and the red line is the spectral response of the filter with an oblique linear amplitude–frequency characteristic. As can be seen from the Figure 3, only high-precision amplitude measurement of the photodetector output current leads to acceptable accuracy of the MAFBS central frequency shift determination. Due to this fact, the subsequent development of microwave-photonic measuring systems based on the generalized modulation factor is a very difficult task, since it requires high accuracy in the amplitude determination of the output current of the photodetector. The relative error of MAFBS central frequency shifting determination does not exceed 10

^{−1}and 10

^{−2}for E

_{F}= 0.01% and 0.01%, respectively. These values cannot be considered acceptable for high-precision measurements.

_{F}= 0.01% (Figure 4a) and E

_{F}= 0.001% (Figure 4b) were made. The relative error of the central frequency shift determination of MAFBS, calculated via Equation (13), does not exceed 10

^{−3}(for E

_{F}= 0.01%) and 10

^{−4}(for E

_{F}= 0.001%), almost in the whole measurement range. The only exception is a small sector, where the amplitudes (A

_{1}, A

_{2}, and A

_{3}) are close to zero.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Amplitude–frequency diagrams: (

**a**) transmitted through Multi-Addressed Fiber Bragg Structure (MAFBS), formed using Fiber Bragg Gratings (FBG) with π-shifts; (

**b**) reflected from MAFBS, formed as a set of ultra-narrowband FBGs.

**Figure 2.**Multi-Addressed Fiber Bragg Structure: (

**a**) a microwave-photonic interrogation scheme; (

**b**) a spectral response of Multi-Addressed Fiber Bragg Structure.

**Figure 3.**Modulation factor as the function of MAFBS central frequency shifting: The blue line is the dependence of the modulation factor; the red line is the spectral characteristic of the optic linear oblique filter, for an amplitude definition accuracy of (

**a**) 0.01% and (

**b**) 0.001%.

**Figure 4.**Relative error of the MAFBS central frequency definition for the amplitude definition error of (

**a**) 0.01% and (

**b**) 0.001%: The thick line is the relative error; the thin lines are the calculated amplitudes of the frequency components.

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**MDPI and ACS Style**

Morozov, O.; Sakhabutdinov, A.; Anfinogentov, V.; Misbakhov, R.; Kuznetsov, A.; Agliullin, T.
Multi-Addressed Fiber Bragg Structures for Microwave-Photonic Sensor Systems. *Sensors* **2020**, *20*, 2693.
https://doi.org/10.3390/s20092693

**AMA Style**

Morozov O, Sakhabutdinov A, Anfinogentov V, Misbakhov R, Kuznetsov A, Agliullin T.
Multi-Addressed Fiber Bragg Structures for Microwave-Photonic Sensor Systems. *Sensors*. 2020; 20(9):2693.
https://doi.org/10.3390/s20092693

**Chicago/Turabian Style**

Morozov, Oleg, Airat Sakhabutdinov, Vladimir Anfinogentov, Rinat Misbakhov, Artem Kuznetsov, and Timur Agliullin.
2020. "Multi-Addressed Fiber Bragg Structures for Microwave-Photonic Sensor Systems" *Sensors* 20, no. 9: 2693.
https://doi.org/10.3390/s20092693